Extensions of Vandermonde Type Convolutions with Several Summations and their Applications - I

Abstract

In an earlier paper [8], one of the authors has established some Vandermonde type convolution identities involving multinomial coefficients with several summations which evidently are regeneralizations of identities in [1] with one summation. In this paper similar identities are derived for coefficients (defined below) of a general type, in the line of the results in [2] and [3], Furthermore, in a series of papers [4], [5], [6], Gould has obtained results on inversion of series and on classical polynomials by an extensive use of these identities with one summation.

On Jensen's and related combinatorial identities

“…, s) and z → −z + 1 in (23) and observing that −x k = (−1) |k| x + |k| − 1 k , we immediately get (19). Comparing (19) with (23) and replacing s with s + 2, we obtain the following result.…”

“…In 1969, Mohanty and Handa [19] established the following multinomial coefficient generalization of Jensen's identity…”

“…Moreover, the identities (1) and (3) were respectively generalized by Mohanty and Handa [19] and Chu [5] to the case of multinomial coefficients (to be stated in Section 4).…”

Bijective proofs of Jensen’s and Mohanty-Handa’s identities

“…This section generalizes the preceding proof of Jensen's identity to a bijective proof of a multivariable generalization of this identity due to Mohanty and Handa [9]. To state the identity, we need some notation.…”

“…This formula has a multivariate generalization called Mohanty-Handa's identity [9], which is stated in equation (3) below. The main purpose of this paper is to provide bijective proofs of these identities based on lattice path models.…”